Intermediate Algebra

1. The Real Numbers

• 1.1 Introduction
  • 1.1.1 An Introduction to Algebra
  • 1.1.2 The Top Ten List of Mistakes

• 1.2 Properties of Real Numbers
  • 1.2.1 Properties, Identities, & Inverses

• 1.3 Inequalities
  • 1.3.1 Concepts of Inequality
  • 1.3.2 Inequalities and Interval Notation

• 1.4 Absolute Value
  • 1.4.1 Properties of Absolute Value
  • 1.4.2 Evaluating Absolute Value Expressions

• 1.5 Operations on Real Numbers
  • 1.5.1 Operations Considering Signs
  • 1.5.2 Prime and Composite Numbers and Their Roots
  • 1.5.3 Order of Operations

• 1.6 Conditional Statements
  • 1.6.1 Forms of Conditional Statements
  • 1.6.2 Inductive Reasoning

2. Equations and Inequalities

• 2.1 Equations in One Variable
  • 2.1.1 An Introduction to Solving Equations
  • 2.1.2 Equality or Identity?
  • 2.1.3 Equivalent Equations and Equations with No Solution
  • 2.1.4 Solving Linear Equations

• 2.2 Applications of Equations Using Formulas
  • 2.2.1 An Introduction to Solving Word Problems
  • 2.2.2 Finding Perimeter
  • 2.2.3 Solving a Linear Geometry Problem
  • 2.2.4 Solving for Constant Velocity

• 2.3 More Applications
  • 2.3.1 Solving a Business Problem
  • 2.3.2 Solving a Mixture Problem
  • 2.3.3 Solving an Investment Problem
  • 2.3.4 Solving for Consecutive Numbers
  • 2.3.5 Finding an Average

• 2.4 Inequalities in One Variable
  • 2.4.1 An Introduction to Solving Inequalities
  • 2.4.2 Solving Inequalities
  • 2.4.3 Solving Word Problems with Inequalities

• 2.5 Compound Inequalities
  • 2.5.1 Sets, Intersections, and Unions
  • 2.5.2 Solving Compound Inequalities
  • 2.5.3 More on Compound Inequalities

• 2.6 Absolute Value Equations
  • 2.6.1 Matching Number Lines with Absolute Value
  • 2.6.2 Solving Absolute Value Equations
  • 2.6.3 Solving Equations with Two Absolute Value Expressions

• 2.7 Absolute Value Inequalities
  • 2.7.1 Solving Absolute Value Inequalities
  • 2.7.2 Solving Absolute Value Inequalities: More Examples

3. Exponents and Polynomials

• 3.1 Understanding Exponents
  • 3.1.1 An Introduction to Exponents
  • 3.1.2 Evaluating Exponential Expressions
  • 3.1.3 Applying the Rules of Exponents
  • 3.1.4 Evaluating Expressions with Negative Exponents

• 3.2 Scientific Notation
  • 3.2.1 Converting between Decimal and Scientific Notation

• 3.3 Polynomial Basics
  • 3.3.1 Determining Components and Degree
  • 3.3.2 Adding and Subtracting Polynomials
  • 3.3.3 Multiplying Polynomials

• 3.4 Techniques for Multiplying Polynomials
  • 3.4.1 The FOIL Method
  • 3.4.2 Multiplying Big Products
  • 3.4.3 Using Special Products

• 3.5 Techniques for Factoring
  • 3.5.1 Factoring Using the Greatest Common Factor
  • 3.5.2 Factoring by Grouping
  • 3.5.3 Factoring Trinomials Completely
  • 3.5.4 Factoring Trinomials: The Guess and Check Method

• 3.6 Special Factoring
  • 3.6.1 Factoring Perfect Square Trinomials
  • 3.6.2 Factoring the Difference of Two Squares
  • 3.6.3 Factoring Sums and Differences of Cubes
  • 3.6.4 Factoring by Any Method

• 3.7 Solving Equations by Factoring
  • 3.7.1 The Zero Factor Property

• 3.8 Division of Polynomials
  • 3.8.1 Using Long Division with Polynomials
  • 3.8.2 Long Division: Another Example

• 3.9 Synthetic Division
  • 3.9.1 Using Synthetic Division with Polynomials
  • 3.9.2 More Synthetic Division

• 3.10 The Remainder Theorem
  • 3.10.1 Using the Remainder Theorem
  • 3.10.2 More on the Remainder Theorem

4. Rational Expressions

• 4.1 The Basics of Rational Expressions
  • 4.1.1 An Introduction to Rational Expressions
  • 4.1.2 Working with Fractions
  • 4.1.3 Writing Rational Expressions in Lowest Terms

• 4.2 Operations with Rationals
  • 4.2.1 Multiplying and Dividing Rational Expressions
  • 4.2.2 Adding and Subtracting Rational Expressions
  • 4.2.3 Rewriting Complex Fractions

• 4.3 Equations with Rationals
  • 4.3.1 Solving a Linear Equation with Rationals
  • 4.3.2 Solving a Linear Equation with Restrictions

• 4.4 Inequalities with Rationals
  • 4.4.1 Solving Rational Inequalities
  • 4.4.2 Solving Rational Inequalities: Another Example
  • 4.4.3 Determining Domain

• 4.5 Applications
  • 4.5.1 Solving a Problem About Work
  • 4.5.2 Resistors in Parallel

• 4.6 Variation
  • 4.6.1 An Introduction to Variation
  • 4.6.2 Direct Proportion
  • 4.6.3 Inverse Proportion
  • 4.6.4 Joint and Combined Proportion

5. Roots and Radicals

• 5.1 Rational Exponents and Radicals
  • 5.1.1 Radical Notation and Properties of Roots
  • 5.1.2 Variables and Negative Values under a Radical
  • 5.1.3 Converting Rational Exponents and Radicals

• 5.2 Simplifying Radical Expressions
  • 5.2.1 Simplifying Radicals
  • 5.2.2 Simplifying Radical Expressions with Variables

• 5.3 Operations with Radical Expressions
  • 5.3.1 Adding and Subtracting Radical Expressions
  • 5.3.2 Rationalizing Denominators

• 5.4 Equations with Radicals
  • 5.4.1 Solving an Equation Containing a Radical
  • 5.4.2 Solving Equations with Two Radical Expressions
  • 5.4.3 Extraneous Roots
  • 5.4.4 Solving an Equation with Rational Exponents

• 5.5 Complex Numbers
  • 5.5.1 Introducing and Writing Complex Numbers
  • 5.5.2 Rewriting Powers of i
  • 5.5.3 Adding and Subtracting Complex Numbers
  • 5.5.4 Multiplying Complex Numbers
  • 5.5.5 Dividing Complex Numbers

• 5.6 Applications
  • 5.6.1 Finding the Length of the Diagonal of a Cube
  • 5.6.2 Finding the Distance and Midpoint between Two Points
  • 5.6.3 Applications in Meteorology

6. Relations and Functions

• 6.1 The Rectangular Coordinate System
  • 6.1.1 Using the Cartesian System
  • 6.1.2 Thinking Visually

• 6.2 An Introduction to Functions
  • 6.2.1 Introducing Relations and Functions
  • 6.2.2 Functions and the Vertical Line Test
  • 6.2.3 Function Notation and Values

• 6.3 Domain and Range
  • 6.3.1 Finding Domain and Range
  • 6.3.2 Domain and Range: An Explicit Example
  • 6.3.3 Satisfying the Domain of a Function
  • 6.3.4 Graphing Important Functions

• 6.4 The Algebra of Functions
  • 6.4.1 Operations on Functions
  • 6.4.2 Composite Functions
  • 6.4.3 Components of Composite Functions

7. The Straight Line

• 7.1 The Slope of a Line
  • 7.1.1 An Introduction to Slope
  • 7.1.2 Finding the Slope Given Two Points

• 7.2 Graphing Linear Equations
  • 7.2.1 Using Intercepts to Graph Lines
  • 7.2.2 Working with Specific Lines
  • 7.2.3 Interpreting Slope from a Graph
  • 7.2.4 Graphing with a Point and the Slope

• 7.3 Linear Equations
  • 7.3.1 Writing an Equation in Slope-Intercept Form
  • 7.3.2 Writing an Equation Given Two Points
  • 7.3.3 Writing an Equation in Point-Slope Form
  • 7.3.4 Matching a Slope-Intercept Equation with its Graph
  • 7.3.5 Slope for Parallel and Perpendicular Lines

• 7.4 Applications of Linear Concepts
  • 7.4.1 Constructing a Linear Model from a Set of Data
  • 7.4.2 Scatterplots and Predictions
  • 7.4.3 Interpreting Line Graphs
  • 7.4.4 Linear Cost and Revenue

8. Systems of Equations

• 8.1 Linear Systems in Two Variables
  • 8.1.1 An Introduction to Linear Systems
  • 8.1.2 Solving a System by Graphing
  • 8.1.3 Solving a System by Substitution
  • 8.1.4 Solving a System by Elimination

• 8.2 Linear Systems in Three Variables
  • 8.2.1 An Introduction to Systems in Three Variables
  • 8.2.2 Solving Systems with Three Variables
  • 8.2.3 Solving Inconsistent Systems
  • 8.2.4 Solving Dependent Systems
  • 8.2.5 Solving Systems with Two Equations

• 8.3 Applications of Linear Systems
  • 8.3.1 Investments
  • 8.3.2 Partial Fractions

• 8.4 Solutions by Matrix Methods
  • 8.4.1 An Introduction to Matrices
  • 8.4.2 Using the Gauss-Jordan Method
  • 8.4.3 Using Gauss-Jordan: Another Example

• 8.5 Determinants
  • 8.5.1 Evaluating 2x2 Determinants
  • 8.5.2 Evaluating nxn Determinants

• 8.6 Cramer's Rule
  • 8.6.1 Using Cramer's Rule
  • 8.6.2 Using Cramer's Rule in a 3x3 Matrix

• 8.7 Working with Inequalities
  • 8.7.1 An Introduction to Graphing Linear Inequalities
  • 8.7.2 Graphing Linear and Nonlinear Inequalities
  • 8.7.3 Graphing the Solution Set of a System of Inequalities

• 8.8 Systems of Nonlinear Equations
  • 8.8.1 Solving a Nonlinear System by Elimination
  • 8.8.2 Solving a Nonlinear System by Substitution

9. Quadratic Equations and Inequalities

• 9.1 The Basics of Quadratics
  • 9.1.1 An Introduction to Quadratics
  • 9.1.2 Solving Quadratics by Factoring

• 9.2 Graphs of Quadratics
  • 9.2.1 Finding x- and y-Intercepts
  • 9.2.2 Nice-Looking Parabolas
  • 9.2.3 Graphing Parabolas

• 9.3 Solving by Completing the Square
  • 9.3.1 Solving by Completing the Square
  • 9.3.2 Completing the Square: Another Example
  • 9.3.3 Finding the Vertex by Completing the Square

• 9.4 Writing Quadratic Equations
  • 9.4.1 Using the Vertex to Write the Equation
  • 9.4.2 Building a Polynomial Equation from Its Solutions

• 9.5 Solving with the Quadratic Formula
  • 9.5.1 Proving the Quadratic Formula
  • 9.5.2 Using the Quadratic Formula
  • 9.5.3 Predicting Types of Solution from the Discriminant
  • 9.5.4 Using the Discriminant to Graph Parabolas

• 9.6 Equations Quadratic in Form
  • 9.6.1 Solving for a Squared Variable
  • 9.6.2 Finding Real Number Restrictions
  • 9.6.3 Solving Fancy Quadratics
  • 9.6.4 Horizontal Parabolas

• 9.7 Formulas and Applications
  • 9.7.1 Solving a Quadratic Geometry Problem
  • 9.7.2 Solving with the Pythagorean Theorem
  • 9.7.3 Solving a Motion Problem
  • 9.7.4 Solving a Projectile Problem
  • 9.7.5 Solving Other Problems

• 9.8 Nonlinear Inequalities
  • 9.8.1 Solving Quadratic Inequalities
  • 9.8.2 Solving Quadratic Inequalities: Another Example

10. Conic Sections

• 10.1 Parabolas
  • 10.1.1 An Introduction to Conic Sections
  • 10.1.2 An Introduction to Parabolas

• 10.2 Equations for Parabolas
  • 10.2.1 Determining Information about a Parabola from the Equation
  • 10.2.2 Writing an Equation for a Parabola

• 10.3 Graphing Parabolas
  • 10.3.1 Shifting Curves along Axes
  • 10.3.2 Shifting or Translating Curves along Axes
  • 10.3.3 Stretching a Graph
  • 10.3.4 Graphing Quadratics Using Patterns

• 10.4 Applications of Parabolas
  • 10.4.1 Finding the Maximum or Minimum of a Quadratic
  • 10.4.2 Maximum Height in the Real World: A Bridge

• 10.5 Circles
  • 10.5.1 The Center-Radius Equation of a Circle
  • 10.5.2 Finding the Center or Radius of a Circle
  • 10.5.3 Decoding the Circle Formula
  • 10.5.4 Solving Circle Problems

• 10.6 Ellipses
  • 10.6.1 An Introduction to Ellipses
  • 10.6.2 Finding the Equation for an Ellipse
  • 10.6.3 Applying Ellipses: Satellites

• 10.7 Hyperbolas
  • 10.7.1 An Introduction to Hyperbolas
  • 10.7.2 Finding the Equation for a Hyperbola
  • 10.7.3 Applying Hyperbolas: Navigation

• 10.8 Identifying Conic Sections
  • 10.8.1 Identifying a Conic
  • 10.8.2 Name That Conic

• 10.9 Square Root Functions
  • 10.9.1 Conic Halves

11. Inverse, Exponential and Logarithmic Functions

• 11.1 Inverse Functions
  • 11.1.1 Understanding Inverse Functions
  • 11.1.2 The Horizontal Line Test
  • 11.1.3 Are Two Functions Inverses of Each Other?
  • 11.1.4 Graphing the Inverse
  • 11.1.5 Finding the Inverse of a Function

• 11.2 Exponential Functions
  • 11.2.1 An Introduction to Exponential Functions
  • 11.2.2 Graphing Exponential Functions: Patterns
  • 11.2.3 Graphing Exponential Functions: More Patterns
  • 11.2.4 The Number e

• 11.3 Using Exponential Functions
  • 11.3.1 Using Properties of Exponents to Solve Exponential Functions
  • 11.3.2 Finding Present and Future Value
  • 11.3.3 Finding an Interest Rate to Match Goals

• 11.4 Logarithmic Functions
  • 11.4.1 An Introduction to Logarithmic Functions
  • 11.4.2 Converting between Exponential and Logarithmic Functions
  • 11.4.3 Graphing Logarithmic Functions
  • 11.4.4 Matching Logarithmic Functions with Their Graphs

• 11.5 Properties of Logarithms
  • 11.5.1 Properties of Logarithms
  • 11.5.2 Expanding a Logarithmic Expression with Properties
  • 11.5.3 Combining Logarithmic Expressions

• 11.6 Evaluating Logarithms
  • 11.6.1 Finding the Value of a Logarithmic Function
  • 11.6.2 Solving for x in Logarithmic Equations
  • 11.6.3 Using the Logarithmic Change of Base Formula
  • 11.6.4 Evaluating Logarithmic Functions with a Calculator

• 11.7 Exponential and Logarithmic Equations
  • 11.7.1 Solving Exponential Equations
  • 11.7.2 Solving Logarithmic Equations
  • 11.7.3 Solving Equations with Logarithmic Exponents

• 11.8 Applications of Exponential and Logarithmic Functions
  • 11.8.1 Compound Interest
  • 11.8.2 Predicting Change

• 11.9 Exponential Growth and Decay
  • 11.9.1 An Introduction to Exponential Growth and Decay
  • 11.9.2 Half Life
  • 11.9.3 Newton's Law of Cooling
  • 11.9.4 Continuously Compounded Interest

12. Further Topics

• 12.1 Sequences and Series
  • 12.1.1 General and Specific Terms
  • 12.1.2 Understanding Sequence Problems
  • 12.1.3 Series Notation, Definitions and Evaluating

• 12.2 Arithmetic Sequences
  • 12.2.1 Finding Terms in Arithmetic Sequences
  • 12.2.2 Finding the Sum of an Arithmetic Sequence

• 12.3 Geometric Sequences
  • 12.3.1 Finding Terms in Geometric Sequences
  • 12.3.2 Finding the Sum of a Geometric Sequence

• 12.4 The Binomial Theorem
  • 12.4.1 Using the Binomial Theorem
  • 12.4.2 Binomial Coefficients